Modeling a curve edge of a plane
The curve modifier is not aimed for this. And Bevel tool doesn't work like this, either. The best way to control planar curvature is by using curves, which may not well benefit topology, however.

I recommend to use Subsurf modifier + edge creases. Add some loop cuts first, then with a few tweaks on the vertex position (G) and adjust crease weight (ShiftE) on some edges, you can make whatever you want with nice topology. I'm sure you also want to bend it later, so topology is important to consider in advance.